From the top of a 7 m high building, the angle of elevation of the top of a cable tower is `60^o`and the angle of depression of its foot is `45^o`. Determine the height of the tower.

Options

A. `7(sqrt2+1)m`
B. `7(sqrt3+1)m`
C. `5(sqrt3+1)m`
D. `13(sqrt3+1)m`

Explanation

To solve the problem step by step, we will follow the given information and apply trigonometric principles.

### Step 1: Draw the Diagram
- Draw a vertical line representing the building, labeled as AB, where AB = 7 m (the height of the building).
- Draw a vertical line representing the cable tower, labeled as AE, where E is the top of the cable tower.
- Draw a horizontal line from point B to point D, where D is the foot of the cable tower.
- The angle of elevation from point A (top of the building) to point E (top of the tower) is 60°.
- The angle of depression from point A to point D (foot of the tower) is 45°.

### Step 2: Identify the Right Triangles
- Triangle AED is formed with angle AED = 60° and triangle ADB with angle ADB = 45°.

### Step 3: Calculate the Distance AD
- In triangle ADB, we can use the tangent function since we have the angle of depression (45°):
\[ \tan(45°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{AD} \]
Here, AB = 7 m, so:
\[ 1 = \frac{7}{AD} \implies AD = 7 \text{ m} \]

### Step 4: Calculate the Height of the Tower (h)
- In triangle AED, we can also use the tangent function:
\[ \tan(60°) = \frac{AE}{AD} \]
Let h be the height of the tower above the building, so AE = 7 + h. Therefore:
\[ \sqrt{3} = \frac{7 + h}{AD} \]
Substituting AD = 7 m:
\[ \sqrt{3} = \frac{7 + h}{7} \]
Cross-multiplying gives:
\[ 7\sqrt{3} = 7 + h \]
Rearranging gives:
\[ h = 7\sqrt{3} - 7 \]

### Step 5: Calculate the Total Height of the Tower
- The total height of the tower (H) is:
\[ H = AB + h = 7 + (7\sqrt{3} - 7) = 7\sqrt{3} \]

### Step 6: Final Expression
- The height of the tower can be expressed as:
\[ H = 7(1 + \sqrt{3}) \]

### Conclusion
The height of the tower is \( 7(1 + \sqrt{3}) \) meters.

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